Hard-Sphere Lattice Gases. II. Plane-Triangular and Three-Dimensional Lattices

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Abstract
Exact low-density and high-density series are derived for the plane-triangular (pt) lattice gas, and in three dimensions, for the simple cubic (sc) and body-centered cubic (bcc) lattice gases, with first-neighbor exclusions. The series are studied by the ratio and Padé approximant extrapolation techniques. In all cases we find a continuous transition to an ordered state at the activities zt=11.05±15 (pt), 1.09±7 (sc) and 0.77±5 (bcc), and the densities ρt/ρ0=0.832±8 (pt), 0.427±20 (sc) and 0.355±20 (bcc). The grand potentials at these points are (pv0/kBT) = Γt=0.839±5 (pt), 0.380±10 (sc) and 0.290±15 (bcc). In all cases the uncertainties refer to the last decimal places. The ordered state is characterized by a long-range order parameter which vanishes continuously and rapidly at the transition point. The transition is signed from above and below by the rapid divergence (faster than a simple pole) of the ``staggered compressibility.'' Apparently, the isothermal compressiibility also diverges to infinity from either side but with considerably less rapidity.

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